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Question: If f(x) = \(\log_{x^{2}}(\mathcal{l}nx)\) then f ′(x) at x = e is...

If f(x) = log⁡x2(lnx)\log_{x^{2}}(\mathcal{l}nx) then f ′(x) at x = e is

A

0

B

1

C

1e\frac{1}{e}

D

12e\frac{1}{2e}

Answer

12e\frac{1}{2e}

Explanation

Solution

f(x) = log⁡x2\log_{x^{2}}{} (lnx) = lnlnxlnx2\frac{\mathcal{l}n\mathcal{l}nx}{\mathcal{l}nx^{2}} = lnlnx2lnx\frac{\mathcal{l}n\mathcal{l}nx}{2\mathcal{l}nx}

f ′(x) = 12[lnxlnx×1x−lnlnx1x](lnx)2\frac{\frac{1}{2}\left\lbrack \frac{\mathcal{l}nx}{\mathcal{l}nx} \times \frac{1}{x}\mathcal{- l}n\mathcal{l}nx\frac{1}{x} \right\rbrack}{(\mathcal{l}nx)^{2}}

f ′(e) = 12(1e−1elnlne)\frac{1}{2}\left( \frac{1}{e} - \frac{1}{e}\mathcal{l}n\mathcal{l}ne \right) = 12e\frac{1}{2e}